Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity
This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integ...
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MDPI AG
2022-03-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/3/141 |
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| author | Dana Bibulova Burkhan Kalimbetov Valeriy Safonov |
| author_facet | Dana Bibulova Burkhan Kalimbetov Valeriy Safonov |
| author_sort | Dana Bibulova |
| collection | DOAJ |
| description | This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work. |
| first_indexed | 2024-03-09T13:51:24Z |
| format | Article |
| id | doaj.art-b59efa5b455a4f19b57d4c348221f5a5 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T13:51:24Z |
| publishDate | 2022-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-b59efa5b455a4f19b57d4c348221f5a52023-11-30T20:50:29ZengMDPI AGAxioms2075-16802022-03-0111314110.3390/axioms11030141Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating InhomogeneityDana Bibulova0Burkhan Kalimbetov1Valeriy Safonov2Department of Higher Mathematics, South Kazakhstan University Named after M. Auezov, Tauke-Khan Ave., 5, Shymkent 160000, KazakhstanDepartment of Mathematics, Akhmed Yassawi University, B. Sattarkhanov 29, Turkestan 161200, KazakhstanDepartment of Higher Mathematics, National Research University «MPEI», Krasnokazarmennaya 14, 111250 Moscow, RussiaThis article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work.https://www.mdpi.com/2075-1680/11/3/141singular perturbationintegro-differential equationrapidly oscillating inhomogeneityregularizationasymptotic convergence |
| spellingShingle | Dana Bibulova Burkhan Kalimbetov Valeriy Safonov Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity Axioms singular perturbation integro-differential equation rapidly oscillating inhomogeneity regularization asymptotic convergence |
| title | Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity |
| title_full | Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity |
| title_fullStr | Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity |
| title_full_unstemmed | Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity |
| title_short | Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity |
| title_sort | regularized asymptotic solutions of a singularly perturbed fredholm equation with a rapidly varying kernel and a rapidly oscillating inhomogeneity |
| topic | singular perturbation integro-differential equation rapidly oscillating inhomogeneity regularization asymptotic convergence |
| url | https://www.mdpi.com/2075-1680/11/3/141 |
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